Harmonic Progression Shortcut, tricks, and tips

Type 1: Find nth term of series a_{n} = \frac{1}{a+(n-1)d}

Question 1 Find the 8^th term in the series \frac{1}{2}, \frac{1}{4}, \frac{1}{6}…….

Options:

A. \frac{1}{20}

B. \frac{1}{14}

C. \frac{1}{16}

D. \frac{1}{18}

Solution:    We know that,
                    {a_{n}} = \frac{1}{a+(n-1)d}

Convert the HP series in AP

We get 2, 4, 6,……

In the given series,

a (first term) = 2

d (common difference) = 2 …. (4 – 2)

Therefore, 8^th term = t₈ = a + (n-1) d

t₈ = 2 + (8– 1) 2

t₈ = 2 + 7 x 2

t₈ = 2 + 14

t₈ = 16

Correct option: C

Question 2  Find the Middle term in this Harmonic series 4 , a , 6

                     Options:

                     A. \frac{22}{5}

                     B. \frac{21}{5}

                     C. \frac{23}{5}

                     D. \frac{24}{5}

Solution:    We know that,
                    {a_{n}} = \frac{1}{a+(n-1)d}

                    we know Inverse of HP is AP.

                    So, \frac{1}{4} , \frac{1}{a} ,\frac{1}{6} is in AP

                    \frac{1}{a} – \frac{1}{4} = \frac{1}{6} – \frac{1}{a}

                    a = \frac{24}{5}

                   Correct option: D